Phonon density of states of a bcc crystal

The form below generates a table of where the first column is the angular frequency ω in rad/s and the second column is the density of states D(ω) in units of s/(rad m³).

D(ω) [10¹⁵ s/(rad m³)]
	ω [10¹² rad/s]

Mass: m =	[10^-24 kg]
Spring constant: C =	[N/m]
Lattice constant: a =	[Ångström]

4.426523e-003	9.760996e+000
1.320022e-002	2.936961e+001
2.197391e-002	4.762874e+001
3.074761e-002	6.618876e+001
3.952130e-002	8.654051e+001
4.829500e-002	1.055701e+002
5.706869e-002	1.231363e+002
6.584238e-002	1.436157e+002
7.461608e-002	1.626818e+002
8.338977e-002	1.819849e+002
9.216347e-002	2.004674e+002
1.009372e-001	2.199757e+002
1.097109e-001	2.369902e+002
1.184846e-001	2.544196e+002
1.272582e-001	2.762941e+002
1.360319e-001	2.945806e+002
1.448056e-001	3.089736e+002
1.535793e-001	3.297813e+002
1.623530e-001	3.464219e+002
1.711267e-001	3.656476e+002
1.799004e-001	3.855343e+002
1.886741e-001	4.070805e+002
1.974478e-001	4.227501e+002
2.062215e-001	4.404849e+002
2.149952e-001	4.578915e+002
2.237689e-001	4.765473e+002
2.325426e-001	4.967121e+002
2.413163e-001	5.120716e+002
2.500900e-001	5.315799e+002
2.588637e-001	5.477282e+002
2.676374e-001	5.672776e+002
2.764111e-001	5.853954e+002
2.851847e-001	6.039144e+002
2.939584e-001	6.183074e+002
3.027321e-001	6.384950e+002
3.115058e-001	6.592479e+002
3.202795e-001	6.756606e+002
3.290532e-001	6.901813e+002
3.378269e-001	7.145952e+002
3.466006e-001	7.290566e+002
3.553743e-001	7.467641e+002
3.641480e-001	7.679957e+002
3.729217e-001	7.854707e+002
3.816954e-001	8.004792e+002
3.904691e-001	8.188478e+002
3.992428e-001	8.355386e+002
4.080165e-001	8.539937e+002
4.167902e-001	8.706526e+002
4.255639e-001	8.925955e+002
4.343376e-001	9.053472e+002
4.431112e-001	9.235836e+002
4.518849e-001	9.394173e+002
4.606586e-001	9.636944e+002
4.694323e-001	9.785615e+002
4.782060e-001	9.943952e+002
4.869797e-001	1.017469e+003
4.957534e-001	1.032578e+003
5.045271e-001	1.054051e+003
5.133008e-001	1.067423e+003
5.220745e-001	1.083297e+003
5.308482e-001	1.099705e+003
5.396219e-001	1.123290e+003
5.483956e-001	1.139885e+003
5.571693e-001	1.157665e+003
5.659430e-001	1.174889e+003
5.747167e-001	1.192816e+003
5.834904e-001	1.214594e+003
5.922641e-001	1.232521e+003
6.010377e-001	1.247598e+003
6.098114e-001	1.261307e+003
6.185851e-001	1.283774e+003
6.273588e-001	1.302703e+003
6.361325e-001	1.322025e+003
6.449062e-001	1.335706e+003
6.536799e-001	1.357079e+003
6.624536e-001	1.377062e+003
6.712273e-001	1.395526e+003
6.800010e-001	1.407316e+003
6.887747e-001	1.431935e+003
6.975484e-001	1.452063e+003
7.063221e-001	1.464131e+003
7.150958e-001	1.484779e+003
7.238695e-001	1.502099e+003
7.326432e-001	1.520527e+003
7.414169e-001	1.534008e+003
7.501906e-001	1.560943e+003
7.589642e-001	1.578377e+003
7.677379e-001	1.595670e+003
7.765116e-001	1.617020e+003
7.852853e-001	1.629950e+003
7.940590e-001	1.651592e+003
8.028327e-001	1.674670e+003
8.116064e-001	1.693727e+003
8.203801e-001	1.713376e+003
8.291538e-001	1.732593e+003
8.379275e-001	1.745190e+003
8.467012e-001	1.772147e+003
8.554749e-001	1.786832e+003
8.642486e-001	1.807266e+003
8.730223e-001	1.830449e+003
8.817960e-001	1.848580e+003
8.905697e-001	1.868759e+003
8.993434e-001	1.888883e+003
9.081171e-001	1.908491e+003
9.168907e-001	1.922214e+003
9.256644e-001	1.951525e+003
9.344381e-001	1.972998e+003
9.432118e-001	1.992538e+003
9.519855e-001	2.019117e+003
9.607592e-001	2.030087e+003
9.695329e-001	2.053338e+003
9.783066e-001	2.078531e+003
9.870803e-001	2.099635e+003
9.958540e-001	2.117498e+003
1.004628e+000	2.137011e+003
1.013401e+000	2.159268e+003
1.022175e+000	2.183413e+003
1.030949e+000	2.207854e+003
1.039722e+000	2.230376e+003
1.048496e+000	2.246351e+003
1.057270e+000	2.266603e+003
1.066044e+000	2.290903e+003
1.074817e+000	2.318522e+003
1.083591e+000	2.335445e+003
1.092365e+000	2.365918e+003
1.101138e+000	2.376185e+003
1.109912e+000	2.403234e+003
1.118686e+000	2.437673e+003
1.127459e+000	2.454633e+003
1.136233e+000	2.388307e+003
1.145007e+000	2.232975e+003
1.153780e+000	2.161566e+003
1.162554e+000	2.108845e+003
1.171328e+000	2.070817e+003
1.180102e+000	2.039150e+003
1.188875e+000	2.006161e+003
1.197649e+000	1.984090e+003
1.206423e+000	1.958915e+003
1.215196e+000	1.940428e+003
1.223970e+000	1.926564e+003
1.232744e+000	1.912471e+003
1.241517e+000	1.889202e+003
1.250291e+000	1.887588e+003
1.259065e+000	1.875060e+003
1.267839e+000	1.870090e+003
1.276612e+000	1.855907e+003
1.285386e+000	1.852902e+003
1.294160e+000	1.842964e+003
1.302933e+000	1.845850e+003
1.311707e+000	1.836385e+003
1.320481e+000	1.830878e+003
1.329254e+000	1.829966e+003
1.338028e+000	1.829473e+003
1.346802e+000	1.828890e+003
1.355575e+000	1.830659e+003
1.364349e+000	1.831370e+003
1.373123e+000	1.837392e+003
1.381897e+000	1.831602e+003
1.390670e+000	1.841094e+003
1.399444e+000	1.839357e+003
1.408218e+000	1.842845e+003
1.416991e+000	1.852095e+003
1.425765e+000	1.854712e+003
1.434539e+000	1.863407e+003
1.443312e+000	1.869270e+003
1.452086e+000	1.879427e+003
1.460860e+000	1.884520e+003
1.469633e+000	1.893702e+003
1.478407e+000	1.902451e+003
1.487181e+000	1.913812e+003
1.495955e+000	1.924831e+003
1.504728e+000	1.938727e+003
1.513502e+000	1.946687e+003
1.522276e+000	1.960383e+003
1.531049e+000	1.979905e+003
1.539823e+000	1.994389e+003
1.548597e+000	2.013213e+003
1.557370e+000	2.031582e+003
1.566144e+000	2.046490e+003
1.574918e+000	2.059201e+003
1.583692e+000	2.087376e+003
1.592465e+000	2.104081e+003
1.601239e+000	2.124163e+003
1.610013e+000	2.151349e+003
1.618786e+000	2.174117e+003
1.627560e+000	2.198144e+003
1.636334e+000	2.224267e+003
1.645107e+000	2.250591e+003
1.653881e+000	2.277458e+003
1.662655e+000	2.313985e+003
1.671428e+000	2.346482e+003
1.680202e+000	2.386406e+003
1.688976e+000	2.418589e+003
1.697750e+000	2.457159e+003
1.706523e+000	2.503821e+003
1.715297e+000	2.543389e+003
1.724071e+000	2.592468e+003
1.732844e+000	2.632401e+003
1.741618e+000	2.696064e+003
1.750392e+000	2.742872e+003
1.759165e+000	2.806608e+003
1.767939e+000	2.878742e+003
1.776713e+000	2.954623e+003
1.785486e+000	3.032643e+003
1.794260e+000	3.122037e+003
1.803034e+000	3.224740e+003
1.811808e+000	3.349927e+003
1.820581e+000	3.483741e+003
1.829355e+000	3.663469e+003
1.838129e+000	3.880112e+003
1.846902e+000	4.218629e+003
1.855676e+000	4.929417e+003
1.864450e+000	5.470092e+003
1.873223e+000	4.391541e+003
1.881997e+000	4.042635e+003
1.890771e+000	3.826265e+003
1.899545e+000	3.660702e+003
1.908318e+000	3.535947e+003
1.917092e+000	3.433609e+003
1.925866e+000	3.353415e+003
1.934639e+000	3.274994e+003
1.943413e+000	3.218061e+003
1.952187e+000	3.160776e+003
1.960960e+000	3.114889e+003
1.969734e+000	3.061875e+003
1.978508e+000	3.027427e+003
1.987281e+000	2.988465e+003
1.996055e+000	2.962214e+003
2.004829e+000	2.924465e+003
2.013603e+000	2.907300e+003
2.022376e+000	2.879485e+003
2.031150e+000	2.864709e+003
2.039924e+000	2.843304e+003
2.048697e+000	2.827849e+003
2.057471e+000	2.808231e+003
2.066245e+000	2.807087e+003
2.075018e+000	2.788121e+003
2.083792e+000	2.777270e+003
2.092566e+000	2.764382e+003
2.101339e+000	2.763018e+003
2.110113e+000	2.762649e+003
2.118887e+000	2.762886e+003
2.127661e+000	2.763269e+003
2.136434e+000	2.770231e+003
2.145208e+000	2.765768e+003
2.153982e+000	2.778282e+003
2.162755e+000	2.786662e+003
2.171529e+000	2.793314e+003
2.180303e+000	2.807593e+003
2.189076e+000	2.826148e+003
2.197850e+000	2.844781e+003
2.206624e+000	2.860191e+003
2.215398e+000	2.889538e+003
2.224171e+000	2.910737e+003
2.232945e+000	2.952298e+003
2.241719e+000	2.985160e+003
2.250492e+000	3.017684e+003
2.259266e+000	3.060148e+003
2.268040e+000	3.106349e+003
2.276813e+000	3.157808e+003
2.285587e+000	3.214796e+003
2.294361e+000	3.284537e+003
2.303134e+000	3.358302e+003
2.311908e+000	3.447018e+003
2.320682e+000	3.551120e+003
2.329456e+000	3.654816e+003
2.338229e+000	3.781481e+003
2.347003e+000	3.945499e+003
2.355777e+000	4.131281e+003
2.364550e+000	4.381137e+003
2.373324e+000	4.889142e+003
2.382098e+000	5.184667e+003
2.390871e+000	5.184147e+003
2.399645e+000	5.176870e+003
2.408419e+000	5.170816e+003
2.417193e+000	5.175708e+003
2.425966e+000	5.184662e+003
2.434740e+000	5.180016e+003
2.443514e+000	5.189039e+003
2.452287e+000	5.182519e+003
2.461061e+000	5.172526e+003
2.469835e+000	5.184202e+003
2.478608e+000	5.197528e+003
2.487382e+000	5.205843e+003
2.496156e+000	5.210129e+003
2.504929e+000	5.223824e+003
2.513703e+000	5.232222e+003
2.522477e+000	5.244445e+003
2.531251e+000	5.262011e+003
2.540024e+000	5.266653e+003
2.548798e+000	5.288117e+003
2.557572e+000	5.304575e+003
2.566345e+000	5.330297e+003
2.575119e+000	5.352217e+003
2.583893e+000	5.374931e+003
2.592666e+000	5.413861e+003
2.601440e+000	5.448710e+003
2.610214e+000	5.493399e+003
2.618987e+000	5.566275e+003
2.627761e+000	5.651945e+003
2.627761	0
3	0

The Matlab file used to calculate the data are bccdos.m.

In the long wavelength limit (near ω = 0) we expect that the density of states should increase like ω². For the case of atoms on a bcc lattice connected to only their nearest neighbors by linear springs, the density of states increases linearly in ω for small ω. If linear springs to next nearest neighbor atoms are included, the density of states increases like ω² as it does for the phonon dispersion curve shown below. The matlab script also handles next nearest neighbors.

The density of states can be used to calculate the temperature dependence of thermodynamic quantities.